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A planar graph is said to be convex if all of its faces (including the outer face) are convex polygons. Not all planar graphs have a convex embedding (e.g. the complete bipartite graph K 2,4). A sufficient condition that a graph can be drawn convexly is that it is a subdivision of a 3-vertex-connected planar graph.
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Every 1-planar graph with n vertices has at most 4n − 8 edges. [4] More strongly, each 1-planar drawing has at most n − 2 crossings; removing one edge from each crossing pair of edges leaves a planar graph, which can have at most 3n − 6 edges, from which the 4n − 8 bound on the number of edges in the original 1-planar graph immediately follows. [5]
In graph theory, Mac Lane's planarity criterion is a characterisation of planar graphs in terms of their cycle spaces, named after Saunders Mac Lane who published it in 1937. It states that a finite undirected graph is planar if and only if the cycle space of the graph (taken modulo 2) has a cycle basis in which each edge of the graph ...
The complement graph of a complete graph is an empty graph. If the edges of a complete graph are each given an orientation, the resulting directed graph is called a tournament. K n can be decomposed into n trees T i such that T i has i vertices. [6] Ringel's conjecture asks if the complete graph K 2n+1 can be decomposed into copies of any tree ...
In an n-vertex connected graph, the largest planar subgraph has at most 3n − 6 edges, and any spanning tree forms a planar subgraph with n − 1 edges. Thus, it is easy to approximate the maximum planar subgraph within an approximation ratio of one-third, simply by finding a spanning tree.
An outerplanar graph (or 1-outerplanar graph) has all of its vertices on the unbounded (outside) face of the graph. A 2-outerplanar graph is a planar graph with the property that, when the vertices on the unbounded face are removed, the remaining vertices all lie on the newly formed unbounded face.
Proof without words that a hypercube graph is non-planar using Kuratowski's or Wagner's theorems and finding either K 5 (top) or K 3,3 (bottom) subgraphs. Wagner published both theorems in 1937, [1] subsequent to the 1930 publication of Kuratowski's theorem, [2] according to which a graph is planar if and only if it does not contain as a subgraph a subdivision of one of the same two forbidden ...